Positive Mean Curvature Research Articles

Mean curvature flow with surgery of mean convex surfaces in $$\mathbb {R}^3$$ R 3

We define a notion of mean curvature flow with surgery for two-dimensional surfaces in $$\mathbb {R}^3$$ with positive mean curvature. Our construction relies on the earlier work of Huisken and Sinestrari in the higher dimensional case. One of the main ingredients in the proof is a new estimate for the inscribed radius established by the first author (Invent Math, 2015).

A holographic principle for the existence of imaginary Killing spinors

Suppose that $\Sigma=\partial\Omega$ is the $n$-dimensional boundary, with positive (inward) mean curvature $H$, of a connected compact $(n+1)$-dimensional Riemannian spin manifold $(\Omega^{n+1},g)$ whose scalar curvature $R\ge -n(n+1)k^2$, for some $k\textgreater{}0$. If $\Sigma$ admits an isometric and isospin immersion $F$ into the hyperbolic space ${\mathbb{H}^{n+1}\_{-k^2}}$, we define a quasi-local mass and prove its positivity as well as the associated rigidity statement. The proof is based on a holographic principle for the existence of an imaginary Killing spinor. For $n=2$, we also show that its limit, for coordinate spheres in an Asymptotically Hyperbolic (AH) manifold, is the mass of the (AH) manifold.

A backward uniqueness result for the wave equation with absorbing boundary conditions

We consider the wave equation $u_{tt}=\Delta u$ on a bounded domain $\Omega\subset{\mathbb R}^n$, $n>1$, with smooth boundary of positive mean curvature. On the boundary, we impose the absorbing boundary condition ${\partial u\over\partial\nu}+u_t=0$. We prove uniqueness of solutions backward in time.

Asymptotic expressions for turbulent burning velocity at the leading edge of a premixed flame brush and their validation by published measurement data

This paper presents validation of new analytical expressions for the turbulent burning velocity, ST, based on asymptotic behavior at the leading edge (LE) in turbulent premixed combustion. Reaction and density variation are assumed to be negligible at the LE to avoid the cold boundary difficulty in the statistically steady state. Good agreement is shown for the slopes, dST/du′, with respect to Lc/δf at low turbulence, with both normalized by those of the reference cases. δf is the inverse of the maximum gradient of reaction progress variable through an unstretched laminar flame, and Lc is the characteristic length scale given as burner diameter or measured integral length scale. Comparison is made for thirty-five datasets involving different fuels, equivalence ratios, H2 fractions in fuel, pressures, and integral length scales from eight references [R. C. Aldredge et al., “Premixed-flame propagation in turbulent Taylor–Couette flow,” Combust. Flame 115, 395 (1998); M. Lawes et al., “The turbulent burning velocity of iso-octane/air mixtures,” Combust. Flame 159, 1949 (2012); H. Kido et al., “Influence of local flame displacement velocity on turbulent burning velocity,” Proc. Combust. Inst. 29, 1855 (2002); J. Wang et al., “Correlation of turbulent burning velocity for syngas/air mixtures at high pressure up to 1.0 MPa,” Exp. Therm. Fluid Sci. 50, 90 (2013); H. Kobayashi et al., “Experimental study on general correlation of turbulent burning velocity at high pressure,” Proc. Combust. Inst. 27, 941 (1998); C. W. Chiu et al., “High-pressure hydrogen/carbon monoxide syngas turbulent burning velocities measured at constant turbulent Reynolds numbers,” Int. J. Hydrogen Energy 37, 10935 (2012); P. Venkateswaran et al., “Pressure and fuel effects on turbulent consumption speeds of H2/CO blends,” Proc. Combust. Inst. 34, 1527 (2013); M. Fairweather et al., “Turbulent burning rates of methane and methane–hydrogen mixtures,” Combust. Flame 156, 780 (2009)]. The turbulent burning velocity is shown to increase as the flamelet thickness, δf, decreases at a high pressure, for an equivalence ratio slightly rich or close to stoichiometric and for mixture of a high H2 fraction. Two constants involved are C to scale turbulent diffusivity as a product of turbulent intensity and characteristic length scale and Cs to relate δf with the mean effective Lm. Lm=Dmu/SLu0 is the scale of exponential decay at the LE of an unstretched laminar flame. The combined constant, KC/Cs, is adjusted to match measured turbulent burning velocities at low turbulence in each of the eight different experimental setups. All measured ST/SLu0 values follow the line, KDtu/Dmu + 1, at low turbulent intensities and show bending below the line due to positive mean curvature and broadened flamelet thickness at high turbulent intensities. Further work is required to determine the constants, Cs and K, and the factor, Lm/Lm*−Lm∇⋅nf, that is responsible for bending in different conditions of laminar flamelet and incoming turbulence.

Geometric flows for quadrature identities

A new geometric flow describing the motion of quadrature surfaces is introduced. This characterization enables us to study quadrature surfaces through the investigation of the flow. It is proved that the flow is uniquely solvable under the geometric condition that the initial surface has positive mean curvature. As a consequence, a bifurcation criterion for quadrature surfaces is obtained. In fact, we study a generalized flow which includes the geometric flow and the Hele-Shaw flow as special cases.

Plateau-Stein manifolds

Abstract We study/construct (proper and non-proper) Morse functions f on complete Riemannian manifolds X such that the hypersurfaces f(x) = t for all −∞ < t < +∞ have positive mean curvatures at all non-critical points x ∈ X of f. We show, for instance, that if X admits no such (not necessarily proper) function, then it contains a (possibly, singular) complete (possibly, compact) minimal hypersurface of finite volume.

Self-shrinkers with a rotational symmetry

In this paper we present a new family of non-compact properly embedded, self-shrinking, asymptotically conical, positive mean curvature ends $\Sigma^n\subseteq\mathbb{R}^{n+1}$ that are hypersurfaces of revolution with circular boundaries. These hypersurface families interpolate between the plane and half-cylinder in $\mathbb{R}^{n+1}$, and any rotationally symmetric self-shrinking non-compact end belongs to our family. The proofs involve the global analysis of a cubic-derivative quasi-linear ODE. We also prove the following classification result: a given complete, embedded, self-shrinking hypersurface of revolution $\Sigma^n$ is either a hyperplane $\mathbb{R}^{n}$, the round cylinder $\mathbb{R}\times S^{n-1}$ of radius $\sqrt{2(n-1)}$, the round sphere $S^n$ of radius $\sqrt{2n}$, or is diffeomorphic to an $S^1\times S^{n-1}$ (i.e. a "doughnut" as in [Ang], which when $n=2$ is a torus). In particular for self-shrinkers there is no direct analogue of the Delaunay unduloid family. The proof of the classification uses translation and rotation of pieces, replacing the method of moving planes in the absence of isometries.

Supersymmetric rigidity of asymptotically locally hyperbolic manifolds

Let (M, g) be an Asymptotically Locally Hyperbolic (ALH) manifold which is the interior of a conformally compact manifold and (∂M, [γ]) its conformal infinity. Suppose that the Ricci tensor of (M, g) dominates that of the hyperbolic space and that its scalar curvature satisfies a certain decay condition at infinity. If the Yamabe invariant of (∂M, [γ]) is non-negative, we prove that there exists a conformal metric on M with non-negative scalar curvature and whose boundary ∂M has either positive or zero constant inner mean curvature. In the spin case, we make use of a previous estimate obtained by X. Zhang and the authors for the Dirac operator of the induced metric on ∂M. As a consequence, we generalize and simplify the proof of the result by Andersson and Dahl in [Scalar curvature rigidity for asymptotically locally hyperbolic manifolds, Ann. Global Anal. Geom.16 (1998) 1–27] about the rigidity of the hyperbolic space when the prescribed conformal infinity ∂M is a round sphere. We also provide non-existence results for conformally compact ALH spin metrics when ∂M is conformal to a Riemannian manifold with special holonomy.

CONTRACTION OF HOROSPHERE-CONVEX HYPERSURFACES BY POWERS OF THE MEAN CURVATURE IN THE HYPERBOLIC SPACE

This paper concerns the evolution of a closed hypersurface of the hyperbolic space, convex by horospheres, in direction of its inner unit normal vector, where the speed equals a positive powerof the positive mean curvature. It is shown that the flow exists on a finite maximal inter- val, convexity by horospheres is preserved and the hypersurfaces shrink down to a single point as the final time is approached.

<i>S</i><sup>1</sup>-Equivariant CMC Surfaces in the Berger Sphere and the Corresponding Lagrangians

The periodic s1-equivariant hypersurfaces of constant mean curvature can be obtained by using the Lagrangians with suitable potential functions in the Berger spheres. In the corresponding Hamiltonian system, the conservation law is effectively applied to the construction of periodic s1-equivariant surfaces of arbitrary positive constant mean curvature.

α-Synuclein Induced Membrane Curvature: What is the Significance of Negative Gaussian Curvature

α-Synuclein (αS) is an intrinsically disordered protein that forms an amphipathic α-helix upon binding to lipid vesicles. Using Molecular Dynamics simulations we have recently shown that the protein induces an anisotropic curvature field in the phospholipid bilayer when bound to a membrane. First, as is known to be characteristic of amphipathic helices, αS induces positive mean curvature (a hill). More interestingly, our simulations suggested unexpected complexities in the induced Gaussian curvature field. Our current efforts, described here, explore the physical principles that dictate these membrane curvature features, again using large-scale simulations. Our focus has been on how three physical aspects of the protein (helix length, flexibility, and hydrophobicity) lead to these complex curvature fields. In addition, we explore the relationship between negative Gaussian curvature and membrane tubulation.

Minimizers of the Willmore functional with a small area constraint

We show the existence of a smooth spherical surface minimizing the Willmore functional subject to an area constraint in a compact Riemannian three-manifold, provided the area is small enough. Moreover, we partially classify complete surfaces of Willmore type with positive mean curvature in Riemannian three-manifolds.

Noncollapsing in mean-convex mean curvature flow

We provide a direct proof of a noncollapsing estimate for compact hypersurfaces with positive mean curvature moving under the mean curvature flow: Precisely, if every point on the initial hypersurface admits an interior sphere with radius inversely proportional to the mean curvature at that point, then this remains true for all positive times in the interval of existence. 53C44; 58J35, 35K93

Mean curvature flow with obstacles

We consider the evolution of fronts by mean curvature in the presence of obstacles. We construct a weak solution to the flow by means of a variational method, corresponding to an implicit time-discretization scheme. Assuming the regularity of the obstacles, in the two-dimensional case we show existence and uniqueness of a regular solution before the onset of singularities. Finally, we discuss an application of this result to the positive mean curvature flow.

α-Synuclein Induces Both Positive Mean Curvature and Negative Gaussian Curvature in Membranes

Using a combination of X-ray scattering, fluorescence correlation spectroscopy, coarse-grained molecular dynamics (MD) simulations and potential of mean force calculations, we have explored the membrane remodeling effects of monomeric α-synuclein (αS). Our initial findings from multiple approaches are that αS (1) causes a significant thinning of the bilayer and (2) stabilizes positive mean curvature, such that the maximum principle curvature matches that of synaptic vesicles, αS-induced tubules, and the synthetic lipid vesicles to which the protein binds most tightly. This suggests that αS binding to synaptic vesicles likely stabilizes their intrinsic curvature. We then show that αS induces local negative Gaussian curvature, an effect that occurs in regions of αS shown previously via NMR and corroborated by MD simulation to have significant conformational flexibility. The induction of negative Gaussian curvature, which has implications for all curvature-sensing and curvature-generating amphipathic α-helices, supports a hypothesis that connects helix insertion to fusion and fission of vesicles, processes that have recently been linked to αS function. Then, in an effort to explain these biophysical properties of αS, we promote an intrinsic curvature-field model that recasts long-range protein-protein interactions in terms of the interactions between the local curvature fields generated by lipid-protein complexes.

Molecular View of the Role of Fusion Peptides in Promoting Positive Membrane Curvature

Fusion peptides are moderately hydrophobic segments of viral and nonviral membrane fusion proteins that enable these proteins to fuse two closely apposed biological membranes. In vitro assays furthermore show that even isolated fusion peptides alone can support membrane fusion in model systems. In addition, the fusion peptides have a distinct effect on the phase diagram of lipid mixtures. Here, we present molecular dynamics simulations investigating the effect of a particular fusion peptide, the influenza hemagglutinin fusion peptide and some of its mutants, on the lipid phase diagram. We detect a systematic shift toward phases with more positive mean curvature in the presence of the peptides, as well as an occurrence of bicontinuous cubic phases, which indicates a stabilization of Gaussian curvature. The wild-type fusion peptide has a stronger effect on the phase behavior as compared to the mutants, which we relate to its boomerang shape. Our results point to a different role of fusion peptides than hitherto assumed, the stabilization of pores rather than stalks along the fusion pathway.

The Nature of Membrane Curvature-Induction by Amphipathic α-Helices Relies upon Protein Length: Simulations of α-Synuclein and H0

Remodeling of membranes, for example the induction of membrane curvature, is a necessary step in vesicle fusion and fission. One mechanism for curvature induction relies upon the interfacial binding of amphipathic proteins, which act as a wedge in one leaflet of the membrane. In this study, we explore differences between the membrane remodeling effects of two biologically important amphipathic proteins, specifically α-Synuclein (αS) and the N-BAR helix H0. αS is an intrinsically disordered protein that adopts an amphipathic α-helical structure upon binding a membrane. Recently it has been shown that αS can induce tubulation and vesiculation of vesicles (Varkey et al. 2010, J Biol Chem, 285(42):32486). In contrast, the amphipathic helix H0 is unable, on its own, to induce tubulation, as N-BAR relies instead upon its large scaffolding domain (Fernandes et al. 2008, Biophys J, 94(8):3065). Using a combination of x-ray scattering and coarse-grained molecular dynamics simulations, we explored the membrane remodeling effects of αS. Our findings suggest that αS 1) causes a significant thinning of the bilayer; and 2) stabilizes an anisotropic curvature-field of both positive mean curvature and negative Gaussian curvature. Simulations of H0 show a similar magnitude of the local curvature as with αS, however owing to its molecular length the curvature-field around H0 is isotropic. We propose that the difference in curvature-field anisotropy explains the difference in the two proteins’ abilities to induce macroscopic curvature (i.e. tubulation) of lipid vesicles.

Critical Points of Wang–Yau Quasi-Local Energy

In this paper, we prove the following theorem regarding the Wang-Yau quasi-local energy of a spacelike two-surface in a spacetime: Let $\Sigma$ be a boundary component of some compact, time-symmetric, spacelike hypersurface $\Omega$ in a time-oriented spacetime $N$ satisfying the dominant energy condition. Suppose the induced metric on $\Sigma$ has positive Gaussian curvature and all boundary components of $\Omega$ have positive mean curvature. Suppose $H \le H_0$ where $H$ is the mean curvature of $\Sigma$ in $\Omega$ and $H_0$ is the mean curvature of $\Sigma$ when isometrically embedded in $R^3$. If $\Omega$ is not isometric to a domain in $R^3$, then 1. the Brown-York mass of $\Sigma$ in $\Omega$ is a strict local minimum of the Wang-Yau quasi-local energy of $\Sigma$, 2. on a small perturbation $\tilde{\Sigma}$ of $\Sigma$ in $N$, there exists a critical point of the Wang-Yau quasi-local energy of $\tilde{\Sigma}$.

Spontaneous instability of soft thin films on curved substrates due to van der Waals interaction

The linear bifurcation theory is used to investigate the stability of soft thin films bonded to curved substrates. It is found that such a film can spontaneously lose its stability due to van der Waals or electrostatic interaction when its thickness reduces to the order of microns or nanometers. We first present the generic method for analyzing the surface stability of a thin film interacting with the substrate and then discuss several important geometric configurations with either a positive or negative mean curvature. The critical conditions for the onset of spontaneous instability in these representative examples are established analytically. Besides the surface energy and Poisson's ratio of the thin film, the curvature of the substrate is demonstrated to have a significant influence on the wrinkling behavior of the film. The results suggest that one may fabricate nanopatterns or enhance the surface stability of soft thin films on curved solid surfaces by modulating the mechanical properties of the films and/or such geometrical properties as film thickness and substrate curvature. This study can also help to understand various phenomena associated with surface instability.

On the characteristic direction of real hypersurfaces in and a symmetry result

Abstract In this paper we show the following property of a non Levi flat real hypersurface in : if the unit characteristic direction T is a geodesic, then it is an eigenvector of the second fundamental form and the relative eigenvalue is constant. As an application we prove a symmetry result of Alexandrov type for compact hypersurfaces in with positive constant Levi mean curvature.